With some abuse of notations, for the approximate reciprocal \(f_n\)  that satisfies Assumption REF , we will denote also \(f_n\)  as its analytic extension to the interior of \(\mathcal {C}_n\) . By Riesz functional calculus (see [1]} and [2]}), we can define
\(\Gamma ^\dagger := f_n(\Gamma )= \frac{1}{2 \pi \iota } \int _{\mathcal {C}_n} (z - \Gamma )^{-1} f_n(z) \,\mathrm {d}z.\) 
